Parental Spending on School-Age Children: Structural Stratification and Parental Expectation

Abstract

As consumption expenditures are increasingly recognized as direct measures of children’s material well-being, they provide new insights into the process of intergenerational transfers from parents to children. Little is known, however, about how parents allocate financial resources to individual children. To fill this gap, we develop a conceptual framework based on stratification theory, human capital theory, and the child-development perspective; exploit unique child-level expenditure data from Child Supplements of the PSID; and employ quantile regression to model the distribution of parental spending on children. Overall, we find strong evidence supporting our hypotheses regarding the effects of socioeconomic status (SES), race, and parental expectation. Our nuanced estimates suggest that (1) parental education, occupation, and family income have differential effects on parental spending, with education being the most influential determinant; (2) net of SES, race continues to be a significant predictor of parental spending on children; and (3) parental expectation plays a crucial role in determining whether parents place a premium on child development in spending and how parents prioritize different categories of spending.

Appendix: Brief Introduction to Quantile Regression Modeling

Quantile regression (QR) modeling is a natural extension of linear regression modeling, providing a fuller characterization of the whole distribution of the response variable, introduced by Koenker and Bassett (1978) and Koenker (2005) and disseminated to social science researchers by Hao and Naiman (2007). The term quantile is denoted by Q ^(p), where 0 < p < 1 indicates the cumulated proportion of population sorted by the response variable (here, consumption expenditure). The QR model offers two advantages. First, it estimates the relationship of a covariate with expenditures at different locations of the expenditure distribution, directly addressing the nature of expenditure inequality. Second, the robust property of quantile regression models is their insensitivity to the top-coding expenditure practice used in surveys including the PSID. In contrast, OLS regression estimates are for the conditional mean of the response variable and are sensitive to the top coding issue.

Let log equivalence-scaled family-shared expenditures or child-specific expenditures be y _i for family i ¹⁰; R _i be a vector of three dummy variables indicating blacks, Hispanics, and other race, with whites being the reference; X _i be a vector of family characteristics; and Z _i be a vector of child characteristics. The full quantile regression model used in this article is shown in Eq. (1):

$$ \begin{array}{c}\hfill {y}_i={\upbeta}_0^{(p)}+{\upbeta}_1^{(p)}{\mathbf{R}}_i+{\upbeta}_2^{(p)}{\mathbf{X}}_i+{\upbeta}_3^{(p)}{\mathbf{Z}}_i+{\upvarepsilon}_i^{(p)},\kern5em \hfill \\ {}\hfill {Q}^{(p)}\left({y}_i\left|{\mathbf{R}}_i,{\mathbf{X}}_i,{\mathbf{Z}}_i\right.\right)={\upbeta}_0^{(p)}+{\upbeta}_1^{(p)}{\mathbf{R}}_i+{\upbeta}_2^{(p)}{\mathbf{X}}_i+{\upbeta}_3^{(p)}{\mathbf{Z}}_i.\hfill \end{array} $$

(1)

The three sets of covariates are entered incrementally. The model is estimated at selected deciles (ps) and four sets of results are reported for four deciles: the first decile, the second decile, the fifth decile (median), and the eighth decile. Unlike OLS estimation based on least squares, the estimation of quantile regression is based on minimizing the weighted absolute distances, with a weight of (1 – p) for data points below a specific quantile and p for data points above the specific quantile. For example, in the quantile regression model at the first decile, families in the lower 10 % are given 0.9 weight, and the upper 90 % the 0.1 weight. Like OLS, quantile regression estimation is based on the whole sample rather than a subsample. We estimate the specified quantile regression models using Stata.

Under the independent and identically distributed (i.i.d.) assumption of errors, asymptotic standard errors (large-sample approximations) can be used to make inferences for quantile regression estimates (which can be obtained from -qreg- in Stata). The i.i.d. error assumption, however, is unlikely to hold. The often-observed skewness and outliers make the error distribution depart from i.i.d. and asymptotic standard errors have been found highly sensitive to minor deviation from the i.i.d. error assumption. Therefore, the bootstrap method should be used. Bootstrap standard errors can be obtained using -bsqreg- in Stata. When we test whether the effect of a covariate differs significantly across quantiles, we need a covariance matrix of the coefficients across quantiles, which can be obtained using the bootstrap approach through a simultaneous quantile regression command -sqreg- in Stata. The recommended number of resamples in bootstrap (reps in Stata) is 50–200. This article reports the results from 50 repetitions because they are similar to the results with 200 repetitions.

The goodness-of-fit statistic for quantile regression is R(p) for pth quantile, analogous to R ² of linear regression models (Koenker and Machado 1999). Like R ², R(p) lies between 0 and 1. Unlike R ², which measures the relative fit of two models in terms of residual variance, R(p) measures the relative fit of two models at a specific quantile in terms of an appropriately weighted sum of absolute residuals. This difference has three implications: (1) although R ² is interpreted as the percentage of variation in the dependent variable explained by the model, R(p) may be interpreted as the percentage of appropriately weighted absolute differences in the dependent variable explained by the model; (2) R ² based on squared differences should be greater than R(p) based on absolute differences; and (3) R(p) constitutes a local measure of goodness of fit for a particular quantile, rather than a global measure of goodness of fit over the entire conditional distribution, like R ².